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A322785
Number of uniform multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.
6
1, 1, 4, 4, 12, 4, 48, 4, 183, 297, 1186, 4, 33950, 4, 139527, 1529608, 4726356, 4, 229255536, 4, 3705777010, 36279746314, 13764663019, 4, 14096735197959, 5194673049514, 7907992957755, 2977586461058927, 13426396910491001, 4, 1350012288268171854, 4, 59487352224070807287
OFFSET
0,3
COMMENTS
A multiset is uniform if all multiplicities are equal. A multiset partition is uniform if all parts have the same size.
LINKS
FORMULA
a(n) = 4 <=> n in { A000040 }. - Alois P. Heinz, Feb 03 2022
EXAMPLE
The a(1) = 1 though a(6) = 48 multiset partitions:
{1} {11} {111} {1111} {11111} {111111}
{12} {123} {1122} {12345} {111222}
{1}{1} {1}{1}{1} {1234} {1}{1}{1}{1}{1} {112233}
{1}{2} {1}{2}{3} {11}{11} {1}{2}{3}{4}{5} {123456}
{11}{22} {111}{111}
{12}{12} {111}{222}
{12}{34} {112}{122}
{13}{24} {112}{233}
{14}{23} {113}{223}
{1}{1}{1}{1} {122}{133}
{1}{1}{2}{2} {123}{123}
{1}{2}{3}{4} {123}{456}
{124}{356}
{125}{346}
{126}{345}
{134}{256}
{135}{246}
{136}{245}
{145}{236}
{146}{235}
{156}{234}
{11}{11}{11}
{11}{12}{22}
{11}{22}{33}
{11}{23}{23}
{12}{12}{12}
{12}{12}{33}
{12}{13}{23}
{12}{34}{56}
{12}{35}{46}
{12}{36}{45}
{13}{13}{22}
{13}{24}{56}
{13}{25}{46}
{13}{26}{45}
{14}{23}{56}
{14}{25}{36}
{14}{26}{35}
{15}{23}{46}
{15}{24}{36}
{15}{26}{34}
{16}{23}{45}
{16}{24}{35}
{16}{25}{34}
{1}{1}{1}{1}{1}{1}
{1}{1}{1}{2}{2}{2}
{1}{1}{2}{2}{3}{3}
{1}{2}{3}{4}{5}{6}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[m], SameQ@@Length/@#&]], {m, Table[Join@@Table[Range[n/d], {d}], {d, Divisors[n]}]}], {n, 8}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 26 2018
EXTENSIONS
More terms from Alois P. Heinz, Jan 30 2019
Terms a(14) and beyond from Andrew Howroyd, Feb 03 2022
STATUS
approved